# Integration: the Area Problem

## Key Questions

• Let us look at the definition of a definite integral below.

Definite Integral
$\setminus {\int}_{a}^{b} f \left(x\right) \mathrm{dx} = {\lim}_{n \to \infty} {\sum}_{i = 1}^{n} f \left(a + i \Delta x\right) \Delta x$,
where $\Delta x = \frac{b - a}{n}$.

If $f \left(x\right) \ge 0$, then the definition essentially is the limit of the sum of the areas of approximating rectangles, so, by design, the definite integral represents the area of the region under the graph of $f \left(x\right)$ above the x-axis.

• If we consider the general integral:

$I = {\int}_{a}^{b} \setminus f \left(x\right) \setminus \mathrm{dx}$

Then the Integration calculates the "net" area between the curve $y = f \left(x\right)$ from $x = a$ to $x = b$, and the $x$-axis.

By "net" area, we consider any area below the curve and above the $x$-axis to be positive , and any area above the curve and below the $x$-axis to be negative.